How many facets does the average polytope with vertices in
have?

Clearly we need to define a probability distribution of
points to answer the question.

Perhaps the most interesting describution for which the answer
is known is the uniform distribution on the unit sphere .
The results of Buchta et al [BMT85] show that the expected number
of facets is
assymtotically with
.
The important fact is that it depends linearly on essentially.
Here the function is defined recursively by

Just to see how large the slope
of this ``linear'' function in is,
we calculate it for :